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High school geometry

Course: High school geometry > Unit 9

Lesson 3: Volume and surface area
Math
High school geometry
Solid geometry
Volume and surface area
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Volume formulas review

Google Classroom
Review the formulas for the volume of prisms, cylinders, pyramids, cones, and spheres.
It may seem at first like there are lots of volume formulas, but many of the formulas share a common structure.

Prisms and prism-like figures

start text, V, o, l, u, m, e, end text, start subscript, start text, p, r, i, s, m, end text, end subscript, equals, left parenthesis, start color #0c7f99, start text, b, a, s, e, space, a, r, e, a, end text, end color #0c7f99, right parenthesis, dot, left parenthesis, start color #ca337c, start text, h, e, i, g, h, t, end text, end color #ca337c, right parenthesis
We always measure the height of a prism perpendicularly to the plane of its base. That's true even when a prism is on it's side or when it tilts (an oblique prism).

Rectangular prisms

Often, we first learn about volume using rectangular prisms (specifically right rectangular prisms), such as by building the prism out of cubes.
Note that any face of a rectangular prism could be its base, as long as we measure the height of the prism perpendicularly to that face.
A cube with a length of l, a width of w, and a height of h.
Volumerectangular prism=(Arearectangle)⋅(height)=((rectangle base)(rectangle height))⋅(prism height)=lwh\begin{aligned} \text{Volume}_{\text{rectangular prism}}&=(\blueE{\text{Area}_{\text{rectangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\left(\blueE{(\text{rectangle base})(\text{rectangle height})}\right)\cdot (\maroonD{\text{prism height}})\\\\ &=\blueE{lw}\maroonD{h} \end{aligned}

Triangular prisms

A triangular prism has a base shaped like a triangle.
A triangular prism with a triangular base of base b, a triangular base of height h, and the length of l.
Volumetriangular prism=(Areatriangle)⋅(height)=(12(triangle base)(triangle height))⋅(prism height)=12bhℓ\begin{aligned} \text{Volume}_{\text{triangular prism}}&=(\blueE{\text{Area}_{\text{triangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\left(\blueE{\dfrac{1}{2}(\text{triangle base})(\text{triangle height})}\right)\cdot (\maroonD{\text{prism height}})\\\\ &=\blueE{\dfrac{1}{2}bh}\maroonD{\ell} \end{aligned}

Cylinders

A circular cylinder is a prism-like figure that has a base shaped like a circle.
A cylinder with a radius r and a height h.
Volumecircular cylinder=(Areacircle)⋅(height)=(π⋅(radius)2)⋅(height)=πr2h\begin{aligned} \text{Volume}_{\text{circular cylinder}}&=(\blueE{\text{Area}_{\text{circle}}})\cdot (\maroonD{\text{height}})\\\\ &=(\blueE{\pi \cdot (\text{radius})^2})\cdot (\maroonD{\text{height}})\\\\ &=\blueE{\pi r^2}\maroonD{h} \end{aligned}

Oblique prisms

In oblique prisms, the bases are in parallel planes,
We still calculate the volume in exactly the same way because of Cavalieri's principle.
Which expression gives the volume of the oblique rectangular prism?
A oblique rectangular prism with its rectangular base with a length of two units and a width of one point five units. The prism's slanted height is five units. Its vertical height is four units.
Choose 1 answer:

Pyramids and pyramid-like figures

start text, V, o, l, u, m, e, end text, start subscript, start text, p, y, r, a, m, i, d, end text, end subscript, equals, start color #7854ab, start fraction, 1, divided by, 3, end fraction, end color #7854ab, left parenthesis, start color #0c7f99, start text, b, a, s, e, space, a, r, e, a, end text, end color #0c7f99, right parenthesis, dot, left parenthesis, start color #ca337c, start text, h, e, i, g, h, t, end text, end color #ca337c, right parenthesis
We also measure the height of a pyramid perpendicularly to the plane of its base. Because of Cavalieri's principle, the same volume formula works for right and oblique pyramid-like figures.

Rectangle-based pyramids

A rectangle-based pyramid has a base shaped like a rectangle.
A rectangular pyramid with the rectangular base's length of l units and width of w units. It has a vertical height of h units.
Volumerectangle-based pyramid=13(Arearectangle)⋅(height)=13((rectangle base)(rectangle height))⋅(pyramid height)=13lwh\begin{aligned} \text{Volume}_{\text{rectangle-based pyramid}}&=\purpleD{\dfrac{1}{3}}(\blueE{\text{Area}_{\text{rectangle}}})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}\left(\blueE{(\text{rectangle base})(\text{rectangle height})}\right)\cdot (\maroonD{\text{pyramid height}})\\\\ &=\purpleD{\dfrac{1}{3}}\blueE{lw}\maroonD{h} \end{aligned}

Cones

A circular cone is a pyramid-like figure that has a base shaped like a circle.
A cone with a radius of r and a vertical height of h.
Volumecircular cone=13(Areacircle)⋅(height)=13(π⋅(radius)2)⋅(height)=13πr2h\begin{aligned} \text{Volume}_{\text{circular cone}}&=\purpleD{\dfrac{1}{3}}(\blueE{\text{Area}_{\text{circle}}})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}(\blueE{\pi \cdot (\text{radius})^2})\cdot (\maroonD{\text{height}})\\\\ &=\purpleD{\dfrac{1}{3}}\blueE{\pi r^2}\maroonD{h} \end{aligned}

Spheres

A sphere with a radius of r.
start text, V, o, l, u, m, e, end text, start subscript, start text, s, p, h, e, r, e, end text, end subscript, equals, start color #a75a05, start fraction, 4, divided by, 3, end fraction, end color #a75a05, pi, left parenthesis, start color #0c7f99, start text, r, a, d, i, u, s, end text, end color #0c7f99, right parenthesis, cubed

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